The Isoperimetric Inequality On The Sphere
Let C be a simple closed curve on a sphere of radius 1. Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that
and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
This inequality was discovered by Paul Lévy (1919) who also extended it to higher dimensions and general surfaces.
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Famous quotes containing the words inequality and/or sphere:
“Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
—Francis Picabia (1878–1953)
“Everything goes, everything comes back; eternally rolls the wheel of being. Everything dies, everything blossoms again; eternally runs the year of being. Everything breaks, everything is joined anew; eternally the same house of being is built. Everything parts, everything greets every other thing again; eternally the ring of being remains faithful to itself. In every Now, being begins; round every Here rolls the sphere There. The center is everywhere. Bent is the path of eternity.”
—Friedrich Nietzsche (1844–1900)